Pendulums and predictability
Before you start
Start with a swing’s repetition, then compare two almost identical double pendulums.
The setupThe simple pendulum has a light string and a bob, with angle measured from downward vertical. The double pendulum has two light rods and equal bobs. Cyan is the experiment; gold is an independent nearby initial condition, not a colliding second object.
What to do and noticeWhy does it accelerate along the arc if the string points toward the pivot? · Does adding mass make this ideal pendulum swing faster? · Will a 0.1° difference remain tiny after 20 seconds?
Predict firstWhy does it accelerate along the arc if the string points toward the pivot?
Start with a swing’s repetition, then compare two almost identical double pendulums.
Prerequisites: The simple pendulum has a light string and a bob, with angle measured from downward vertical. The double pendulum has two light rods and equal bobs. Cyan is the experiment; gold is an independent nearby initial condition, not a colliding second object.What brings the bob back?
Predict firstWhy does it accelerate along the arc if the string points toward the pivot?
Gravity is always downward; its tangential component −mg sinθ restores the bob toward the bottom. Radial tension constrains its path. At the bottom T = mg + mv²/L, greater than weight, providing centripetal acceleration. T is not always mg. Red marks weight, gold tension and cyan velocity.
How to observe
- Why does it accelerate along the arc if the string points toward the pivot?
- Does adding mass make this ideal pendulum swing faster?
- Will a 0.1° difference remain tiny after 20 seconds?
Model notes
Point masses without friction, g = 9.81 m/s². Simple release angles stay ≤75° to keep the string taut; double pendulums use rods that also sustain compression. Nonlinear equations are numerically integrated. Not all double-pendulum conditions are strongly chaotic; small motions are more regular.